Can blender create images to look like vector graphics?
Yes, you definitely can - but it's not a slider or a checkbox that will instantly do this for you.

I suggest getting familiar with the Blender Internal Renderer, everything you need to achieve the style you have linked to on your page, you will find in there.
(You can just as well use Cycles too and achieve such styles I guess, but I think BI is the more economic choice here.)

Although it's been suggested above, I don't see any need to look into Freestyle Rendering (essentially an optional line rendering functionality subset of BI, and from the next release on Cycles as well) - after all you hardly have any edges in your illustration style.

Just to give you some pointers for your path from here: You will want to look into shadeless materials (this is a simple checkbox, in fact :)) and familiarize yourself with the material node system, I'd think the Normal node will be involved with some of the techniques that could work for your style. Check out the Blend texture (essentially: Gradient), this will be a good friend on the way. I think the flat, shadeless shadows might be a task for the compositor, e.g. flattening out a shadow pass with colour correction. Experiment! In any case, you will mostly have to figure this out yourself, which will be some work, but will gain you so much, so be fearless und jump right at it. ;) If you have specific problems feel free to look around here or open up new questions if something hasn't been asked and answered previously. Good luck!